Let $d\ge1$ be an integer, $W_d$ be the Witt  algebra. For any admissible $W_d$-module $P$ and any $gl_d$-module $V$, one can form a $W_d$-module $F(P,V)$, which as a vector space is $P\otimes V$.

Since $W_d$ has a natural subalgebra isomorphic to $sl_{d+1}$, we can view $F(P,V)$ as an $sl_{d+1}$-module. Taking $P=\Omega(\bf{\lambda})$, the rank-$1$ $U(\mathfrak{h})$-free $W_d$-module and $V=V(\bf{a},b)$, the irreducible cuspidal module over $gl_d$, we get the special $sl_{d+1}$-module $F(\bf{\lambda}; \bf{a},b)=F(\Omega(\bf{\lambda}),V(\bf{a},b))$. We determine the necessary and sufficient conditions for the $sl_{d+1}$-module $F(\bf{\lambda};\bf{a},b)$ to be irreducible. And for the reducible case, we constructed their proper submodules explicitly.